The first objective in the first step of the three-step process was the approximation of temperatures within the frames. A thermal finite element analysis was performed for each examined load case to approximate a steady-state thermal gradient through the cross sections of the frames. The frame was meshed into 3D thermal elements. The elements were assigned conductivity, and temperatures were applied as boundary conditions in external nodes on the inner and outer surface of the frame.
A steady-state thermal gradient is a temperature distribution, which is independent of time. The distribution of temperatures was approximated based on Fourier’s law of heat flow. This law
Element response
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states that heat will flow in the system if temperature differences are present. The steady-state gradient was then calculated by assuming that energy was conserved in the system (Hens 2010).
In this thermal finite element analysis, effects caused by convection, internal heat generation and radiation are neglected. These effects could have been taken into account by an extended analysis, but in this case, surface temperatures from a steady-state situation were given (Vecchio & Sato 1990). Because of this, it is assumed that the difference between these types of analyses is small in this particular case.
3.3.1 Thermal element response
The experimental geometry of the concrete was meshed into SOLID70 elements from the ANSYS element library. This element was used since the nodal temperature solution was compatible with the SOLID185 element used in the structural analysis. The SOLID70 and SOLID185 element were equal with respect to size, form and number of nodes. The size and form used for the elements were chosen due to approximation of deformations.
The SOLID70 element had eight external nodes with temperature as the only degree of freedom in each node. The distribution of temperatures within the element is approximated based on interpolation functions. These functions are adapted to the geometry of the element by a polynomial expression (ANSYS® Academic Research Mechanical Release 18.0). The element is illustrated in Figure 9.
Figure 9: Eight-node thermal element used for thermal finite element analysis.
The heat flow within the elements is three dimensional, but occurs only in the directions where temperature difference is present. This is according to Fourier’s law of heat flow. This law is the basis for a thermal finite element analysis (Huebner et al. 2001). Fourier’s law is applied together with the law of conservation of energy to approximate a steady-state solution for the elements. Energy conservation ensures that the amount of heat within the element remains
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constant (Hens 2010; Huebner et al. 2001). This implies that heat flowing from one node in the element needs to be received by the other nodes in the element.
The heat flow within the element is approximated with the vector of nodal temperatures and a matrix describing the conductance of the material. The derivation of this conductance matrix for an element is shown in Eq. (3) (Huebner et al. 2001). This equation approximates the conductance related to conduction with the interpolation functions and the conductivity of the material.
K ( )= B k B dΩ
( ) (3)
The matrices used in Eq. (3) are:
K ( ) Element conductance matrix related to conduction.
B Temperature gradient interpolation matrix.
k Thermal conductivity matrix.
Ω( ) Volume of element.
The only material property given as input in the calculation of the conductance matrix related to conduction was the conductivity of the concrete. This property was calculated from a given thermal diffusivity; see Sec. 2.3. The equation used for calculation of the conductivity is given in Eq. (4) (Byron Bird et al. 2002). The conductivity of the concrete was applied equal in all the directions of the material.
k = α ∙ ρ ∙ c (4)
The factors used in Eq. (4) are:
k Conductivity (W/m·°C).
α Thermal diffusivity (m2/s) . ρ Density (kg/m3).
c Specific heat capacity (J/kg·°C).
The density and specific heat capacity were needed to calculate the conductivity from thermal diffusivity. The density was set equal to 2400 kg/m3 by assuming normal concrete (CEN 2002a). The specific heat capacity of concrete was assumed constant and equal to 1000 J/kg·°C for the examined load cases (Klieger & Lamond 1994).
3.3.2 Thermal system response
The local responses of the elements were assembled into a global system response. This was done by adding the local conductance properties into their respective position in the global conductance matrix. The global conductance matrix made the nodal temperatures dependent on the heat flow through all elements connected to the node. In this way, it provide continuity of temperatures within the frame. The general equation for the system is given in Eq. (5) (Huebner et al. 2001).
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K T = R (5)
The matrices and vectors used in Eq. (5) are:
K Global conductance matrix related to conduction (W/°C).
T Nodal temperature vector (°C).
R Nodal heat load vector from specified nodal temperatures (W).
3.3.3 Solution of thermal finite element analysis
The general equation is solved by setting either a temperature or a heat load for each node in the system. This was done in the project, by application of temperatures on the inner and outer surface of the frame. The temperatures used on these surfaces were taken from the article by Vecchio and Sato (1990); see Sec. 2.4. These temperatures were changed for each load case according to the given data.
The nodes on the inner surface of the frame were given high temperatures and the nodes on the outer surface were given low temperatures. Temperatures were applied in all nodes on the respective surfaces. This is a simplification of the real temperature distribution since the measured temperatures were from a region with water on the inside. The simplification was performed since no information was given about conditions above the water surface.
The temperature difference between the two surfaces generates heat flows in the system. The heat flow between the surfaces is illustrated in Figure 10.
Figure 10: Heat flow between warm and cold surface.
The heat flow illustrated in Figure 10 is one-dimensional, but the nodes in the area around a corner would experience heat flow in two dimensions. This effect is caused by cooling from to surfaces.
In the nodes where temperatures are unknown, the heat load is set to zero. The calculation of an unknown temperature is then based on equilibrium between heats flowing in and out of the node due to the defined temperatures.
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